Research article

Sang Hyun Parka, Sung-Gyu Leea, Soojeong Baek, Taewoo Ha, Sanghyub Lee, Bumki Min, Shuang Zhang, Mark Lawrence* and Teun-Teun Kim*

Observation of an exceptional point in a non-Hermitian metasurfacehttps://doi.org/10.1515/nanoph-2019-0489Received November 29, 2019; revised January 19, 2020; accepted January 20, 2020

Abstract: Exceptional points (EPs), also known as non-Hermitian degeneracies, have been observed in parity-time symmetric metasurfaces as parity-time symmetry breaking points. However, the parity-time symmetry condition puts constraints on the metasurface parameter space, excluding the full examination of unique proper-ties that stem from an EP. Here, we thus design a general non-Hermitian metasurface with a unit cell containing two orthogonally oriented split-ring resonators (SRRs) with overlapping resonance but different scattering rates and radiation efficiencies. Such a design grants us full access to the parameter space around the EP. The para-meter space around the EP is first examined by varying the incident radiation frequency and coupling between SRRs. We further demonstrate that the EP is also observ-able by varying the incident radiation frequency along

with the incident angle. Through both methods, we val-idate the existence of an EP by observing unique level crossing behavior, eigenstate swapping under encircle-ment, and asymmetric transmission of circularly polar-ized light.

Keywords: metamaterials; non-Hermitian photonics; exceptional point; asymmetric polarization conversion; phase singularity.

1 IntroductionWhen studying the physics of closed systems in which the conservation of energy is strictly obeyed, Hamiltoni-ans are assumed to be Hermitian due to the many appro-priate mathematical constraints it puts on the system, such as realness of eigenvalues, existence of an ortho-normal basis, and unitary time evolution. Many systems in physics are, however, conveniently described by open systems, which may either gain or lose energy through interactions with its environment, resulting in complex eigenenergies. Such open systems are represented by non-Hermitian Hamiltonians.

One of the most interesting aspects of non-Hermi-tian (open) systems is their degeneracies, also known as exceptional points (EPs), in which both complex eigen-values and eigenvectors of the system coalesce to form a defective eigenspace [1–4]. Unlike the Dirac point, which is unstable under perturbations in two dimensions, the EP is topologically robust, making it an ubiquitous feature among non-Hermitian systems. The significance of EPs has frequently been noted in non-Hermitian systems with special constraint of being parity-time symmetric. In these parity-time symmetric systems, the EP is shown to be the point of the parity-time symmetry breaking transition [5–12]. Beyond being the phase tran-sition point of parity-time symmetric systems, EPs have unique properties due to the underlying geometry of their complex parameter spaces. Mathematically, the complex parameter space forms a self-intersecting Riemann sheet

aSang Hyun Park and Sung-Gyu Lee: These authors contributed equally to this work.*Corresponding authors: Mark Lawrence, Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA, e-mail: markl89@stanford.edu; and Teun-Teun Kim, Center for Integrated Nanostructure Physics (CINAP), Institute for Basic Science (IBS), Suwon 16419, Republic of Korea; and Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea, e-mail: t.kim@skku.edu. https://orcid.org/0000-0002-4154-9775 Sang Hyun Park: Center for Integrated Nanostructure Physics (CINAP), Institute for Basic Science (IBS), Suwon 16419, Republic of KoreaSung-Gyu Lee, Taewoo Ha and Sanghyub Lee: Center for Integrated Nanostructure Physics (CINAP), Institute for Basic Science (IBS), Suwon 16419, Republic of Korea; and Department of Energy Science, Sungkyunkwan University, Suwon 16419, Republic of Korea. https://orcid.org/0000-0003-3757-5863 (S.-G. Lee)Soojeong Baek and Bumki Min: Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Republic of KoreaShuang Zhang: School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom

Open Access. © 2020 Mark Lawrence, Teun-Teun Kim et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License.

Nanophotonics 2020; 9(5): 1031–1039

in which the point of intersection is the EP. The nontriv-ial topology in the parameter space gives rise to intrigu-ing phenomena, such as unconventional level crossing behavior [13, 14], increased sensitivity to perturbations [15–18], and observation of geometric phase under encir-clement [19–23]. Such behavior has been observed using microwave cavities, chaotic optical cavities, and exciton-polariton billiards.

Metasurfaces, two-dimensional arrays of artificially designed subwavelength structures, provide an interest-ing platform for exploring the physics of EPs. The ability to tune the transmission needed to probe the parameter space in the vicinity of an EP is readily provided in meta-surfaces by the variability of array parameters or input frequency. Moreover, measuring the transmission of the metasurface yields “complex” transmission eigenvalues, which is crucial to observing the properties of the EP and non-Hermitian matrices. Indeed, EPs have been observed as a phase transition point in parity-time symmetric meta-surfaces [9, 10]. Although the constraints of parity-time symmetry provide a straightforward path for navigating directly through the EP, they also restrict us from explor-ing the full parameter space around the EP.

In this paper, we demonstrate that topologically robust EPs and phase singularities can be observed in a non-Hermitian metasurface. By designing a meta-surface composed of hybrid meta-atoms with aniso-tropic radiation loss, the polarization eigenstates of the metasurface can be manipulated through variation of radiation frequency and meta-atom separation. By ana-lyzing the transmission of the metasurface, we observe phenomena unique to EPs, including unconventional level crossing behavior, eigenstate swapping under

encirclement of the EP, and asymmetric transmission of circularly polarized radiation. We have also observed a polarization phase singularity for the first time at an EP in the transmission through an anisotropic metasurface. The increased sensitivity to perturbations near the EP implies that the abrupt phase change observed at the singularity may be used in sensing applications [24–26]. Finally, we demonstrate that the EP can be observed without repeated fabrication of samples by simply altering the incident angle of radiation. This method provides a path to observing phenomena related to the dynamic encircling of the EP [27].

2 Results and discussion

2.1 Unit cell design

In this section, we discuss the design of the metasurface unit cell and how it gives a non-Hermitian transmission matrix. The metasurface is composed of an array with two orthogonally oriented split-ring resonators (SRRs) in each unit cell (Figure 1A). The two SRRs in a unit cell are modeled as an effective dipole moment , , ,i t

x y x yp p e ω= which couple strongly to an incident radiation field i t

i iE E e ω= with a radiative coupling strength gi. The geometry and material of the SRR determine the resonance frequency, damping coefficient, and radiative coupling strength of the effec-tive dipole moments. For incident radiation that is close to resonance (δ = ω – ω0 ≈ 0) and assuming small damping (γ  ω0), the governing equation for the dipole moments is given by

S

A B

Tra

nsm

issi

on

0.35 0.40 0.45 0.50 0.550.0

0.2

0.4

0.6

0.8

1.0

Frequency (THz)

100 µm

CtSi

L2

d1L1

L3 d2

L4

Figure 1: Structure of the non-Hermitian metasurface with the transmission spectrum of its resonator components.(A) Schematic of the unit cell geometry. Dimensions are set to L1 = L3 = 30, L2 = 55, L4 = 80, d1 = 10, d2 = 50, tSi = 525. All units are in μm. The dimensions of each SRR are optimized such that they have an equal resonance frequency while having different gap sizes. The difference in gap size makes the transmission matrix dispersive, thus letting us control the eigenstates of the system by changing the frequency of incident radiation. (B) Independent transmission plots for the orthogonally oriented SRRs in the unit cell. Each SRR was excited by incident radiation polarized parallel to the gap direction. (C) Top-view microscopic image of the fabricated metasurface.

1032 S.H. Park et al.: Observation of an exceptional point in a non-Hermitian metasurface

0,

0xx xyx x x x ix

y y y y iyyx yy

G Gi p g Ei p g EG G

δ γ

δ γ

+ + = +

(1)

where Gij is a summation of the i-oriented radiation fields formed by the j-oriented dipole moments. As the cou-pling terms Gij depend only on the lattice geometry, the use of a square unit cell enforces the conditions Gxx = Gyy and Gxy = Gyx. Furthermore, as the dipoles are orthogonally oriented and in the same plane, it follows that coupling between the dipoles only occurs via near-field quasi-static radiation, resulting in a real-valued Gxy. Although aligning the resonance frequencies of the SRRs is not a necessary condition for observing the EP, to maximize the coupling between the two components, the resonance frequencies are aligned (Figure 1B).

The transmitted field can then be found by super-posing the incident field and the field radiated in the forward direction due to the oscillating dipoles, given

by 02 ( , ),

2 x x y y

ig p g p

aωη

where 0 0 0/η µ ε= is the free space

impedance and a is the lattice constant. The transmission matrix is then

2

2

( ),

( )x y xx x y xy

x y xy y x xx

g i G g g GT

g g G g i Gδ γ

δ γ

+ + −= − + +

(2)

up to a term proportional to the identity matrix that will have no effect on the eigenstates of the system. As the transmission matrix is non-Hermitian, we expect to observe non-Hermitian phenomena in the transmission eigenvalues and eigenstates. The eigenstates are Jones vectors that represent polarization states, and the corre-sponding eigenvalues will give the magnitude and phase with which the eigenstates are transmitted. Note that if gx = gy = g, Eq. (2) can be rewritten as

2 1 0( )

0 1xy

xxxy

i Gg i G

G iγ

δ γγ

− + + + − −

(3)

where γ ̅ = (γ1 + γ2)/2, γ = (γ1–γ2)/2. As the identity matrix has no effect on the eigenstates, the matrix eigenstates become dispersionless, with the eigenvalues only linearly depending on the incident radiation frequency. Moreover, we notice that the second term in the brackets is a parity-time symmetric matrix, indicating that we will observe the parity-time symmetric phase transition in the eigenval-ues and eigenvectors of T. Thus, it is important to ensure that gx ≠ gy, which is achieved by making the gaps of two orthogonally oriented SRRs different.

2.2 Methods

The system is first studied numerically using the commer-cial finite-element method solver CST Microwave Studio. We check to find a geometry that gives resonant frequen-cies in the terahertz (THz) range. The unit cell geometry of the metasurface used in the simulations and in subse-quent experiments is shown in Figure 1A, with the reso-nant frequency of each SRR shown in Figure 1B.

The fabrication procedure of metasurfaces is achieved by conventional photolithography. A positive photoresist [AZ-GXR-601 (14cp); AZ Electronic Materials] is spin coated onto a silicon substrate with resistivity of ~10 W cm. Au/Cr (100/5 nm) is then evaporated and patterned as orthogo-nally oriented SRR array structures via a lift-off process.

Transmission spectra of the metasurface are char-acterized by THz time-domain spectroscopy (THz-TDS). THz-TDS yields both the magnitude and phase of the transmission spectrum in a single measurement, giving us easy access to the complex transmission eigenvalues of the metasurface. The THz signal is emitted by a commer-cial photoconductive antenna (iPCA, BATOP). An electro-optic sampling method with a 1-mm-thick ZnTe crystal is used to detect the transmitted THz signals. After taking the Fourier transform of the time signal and normalizing against transmission through the bare silicon substrate, this measurement results in a complex transmission spec-trum characterizing the sample. The full 2 × 2 transmission matrix is measured on the linear polarization basis. The polarization of the incident and transmitted fields are fixed using two wire-grid polarizers on each side of the metasur-face. Transmission for circularly polarized states is related to the transmission of linearly polarized states by

( ) ( ) ( ) ( )1( ) ( ) ( ) ( )2

LL LRcir

RL RR

xx yy xy yx xx yy xy yx

xx yy xy yx xx yy xy yx

T TT

T TT T i T T T T i T TT T i T T T T i T T

=

+ + − − − +

= − + + + − − (4)

where Tij indicates a transmitted wave with polarization i and an incident wave with polarization j.

2.3 Results

2.3.1 Eigenvalue surface topology

Here, we validate the existence of an EP by examining the transmission eigenvalue surface. As mentioned previ-ously, the eigenvalue surface of a non-Hermitian matrix

S.H. Park et al.: Observation of an exceptional point in a non-Hermitian metasurface 1033

forms a self-intersecting Riemann sheet in which the EP is located at the point of intersection (see Figure 2A). This type of topology is unique to non-Hermitian systems and not observed in Hermitian systems. For Hermitian systems, the eigenvalue surface forms a double-cone structure with the degeneracy located at the conical intersection. Hence, observing the self-intersecting Riemann surface structure will provide strong evidence for the observation of an EP.

A non-Hermitian degeneracy has a codimension of 2, meaning that we need two parameters to capture the degeneracy [2]. Based on the ease and independence with which they can be controlled, parameters ω and Gxy are chosen. ω is directly controlled through incident radiation

and Gxy is controlled by changing the distance between SRRs in the unit cell [9, 10]. Computing the eigenvalues and eigenvectors of the transmission matrix given in Eq. (2),

we find that EPs exist at 2 2

EP EP0( , ) , .

2x y y x

xyx y

g gG

g gγ γ

ω ω −

= ±

When calculating the position of the EP, we assume that Gxx is negligible. This is because the largest contribution to Gxx comes from dipoles in neighboring unit cells, whereas Gxy has contributions from within the unit cell. The eigenstate at each EP is (1, ) / 2,i± which corresponds to left and right circularly polarized light. In this paper, we focus on the EP with left circularly polarized light as its singular eigenstate.

Frequency (THz) Frequency (THz)

Eig

entr

ansm

issi

on m

agni

tude

Eig

entr

ansm

issi

on p

hase

Magnitude Phase

ω ω

• EPEP

5 6

S (µm) 5 6

S (µm)

0.455 0.460 0.465 0.455 0.460 0.4650.5

0.6

0.7

0.8

–1.0

–0.5

0.0

0.5

1.0

0.2

0.4

0.6

0.8

1.0

Exp. Exp.

Sim. Sim.

•

0.425 0.430 0.435 0.440 0.425 0.430 0.435 0.4400.5

0.6

0.7

0.8

Gxy

A

B

Gxy

Figure 2: Eigenvalue surface topology near an exceptional point revealed through transmission simulation and experiment results of the non-Hermitian metasurface.(A) Schematic of the complex eigenvalue surface in [ω, Gxy(S)] parameter space. At a given value of Gxy, we see that the crossing behavior as a function of frequency for the magnitude and phase of the eigenvalues is opposite. As the coupling is varied through the EP, the crossing behavior of the magnitude and phase are interchanged. Such behavior is a result of the unique topology of the self-intersecting Riemann surface formed in the vicinity of an EP. Purple lines show the evolution of the eigenvalues as the EP is encircled. (B) Simulation and experimental results of the complex eigenvalues as a function of frequency conducted at different separation S are shown. When S = 5 μm, we see that the magnitude the eigenvalues do not intersect while the phase crosses. When S is changed to 6 μm, the crossing behavior is swapped, and we see crossing in the magnitude and anti-crossing in the phase. As graphically shown in (A), this indicates that an EP exists between S = 5 μm and S = 6 μm.

1034 S.H. Park et al.: Observation of an exceptional point in a non-Hermitian metasurface

The topology of the self-intersecting Riemann surface can be captured by observing the level crossing behavior in the magnitude and phase of the eigenvalues [13, 14].

Although it is usual to examine the real and imaginary parts of the eigenvalues, these quantities lack a direct physical interpretation for the complex transmission

LCPA

B

Frequency (THz)

|T

|R

L

0.01

0.1

1

1

0.01

0.1

5 6 7 8

S (µm)

5 6 7 8

S (µm)

–8

–6

–4

–2

2

–6

–4

–2

0

2

Frequency (THz)

Frequency (THz)

Pha

se T

RL

(rad

.)

LCP

RCP

C

Exp.

Sim.

Exp.

Sim.

S (µm)

6 5

0.35 0.40 0.45 0.50 0.550.001

0.01

0.1

1

T RL,LL

T RL

T LR

Sim. Exp.

Tra

nsm

issi

on

0.420.40 0.44 0.46 0.48 0.50 0.52 0.420.40 0.44 0.46 0.48 0.50 0.52

0.38 0.40 0.42 0.44 0.46 0.48 0.500.38 0.40 0.42 0.44 0.46 0.48 0.50

EP

Sim. Exp.

Start

LCP

RCP

ωGxy

Figure 3: Eigenstate swapping and transmission phase singularity observed by changing coupling between resonator components.(A) Transmission eigenstates of the metasurface are plotted on a Poincaré sphere as the EP is encircled in the [ω, Gxy(S)] parameter space. Under one full loop around the EP, the states move halfway around the north pole of the Poincaré sphere, resulting in a swapping of eigenstates [i.e. (ψ1,ψ2) → (ψ2,-ψ1)]. To the right of the Poincaré sphere, we explicitly show the evolution of polarization states. The states in the dashed box show the two eigenstates at the beginning of the loop. Thereafter, we follow how state ψ1 evolves and find that it becomes ψ2 at the end of a single loop. (B) The transmission matrix components in the circularly polarized basis are shown when S = 6 μm. As we have anticipated from Eq. (5), TRR and TLL are identical, whereas TRL is suppressed with respect to TLR. The asymmetric transmission of circularly polarized light is a direct consequence of having a single eigenstate at the EP. (C) An abrupt phase flip in TRL is observed as the parameter S is modulated through the EP.

S.H. Park et al.: Observation of an exceptional point in a non-Hermitian metasurface 1035

eigenvalues. Therefore, we instead examine the magni-tude and phase of the eigenvalues. As using the magni-tude and phase instead of the real and imaginary parts of the eigenvalues can be considered as a mapping from Cartesian to polar coordinates in the complex plane, the topology near EPs will be preserved. When the value of Gxy is smaller than EP ,xyG the magnitude (phase) of the eigen-values anti-cross (cross). As Gxy is increased and passes through the EP, the crossing behavior interchanges such that the magnitude (phase) now cross (anti-cross). This type of crossing behavior is unique to non-Hermitian systems and cannot be observed in the double-cone eigen-value surface of Hermitian systems.

The transmission eigenvalues calculated from simu-lation and experimental results are shown in Figure 2B. Assuming that the SRR separation distance S is inversely proportional to the coupling strength Gxy, we observe that the magnitude (phase) of the transmission eigenvalues change from anti-crossing (crossing) to crossing (anti-crossing) as coupling strength is increased. Such behav-ior exactly follows the topology of the self-intersecting Riemann surface and provides strong evidence that an EP is located in the parameter range 5 < SEP < 6.

2.3.2 Eigenstate swapping

The eigenvalue surface structure shown schematically in Figure 2A also reveals that the EP is a square-root branch point for the complex eigenvalues. This implies that the eigenvalues of the system will not return to their original values under one full loop around the EP but will rather be swapped. They will return to their original values only under two full loops. For the eigenstates, the EP is in fact a fourth-root branch point, meaning that four full loops are required to return the eigenstates to their initial states [2]. A single loop around the EP will interchange the two eigenstates and additionally accumulate a geometric phase of π on one of the states.

To show the swapping of eigenstates under encircle-ment, we follow the evolution of the transmission eigen-states by observing closely spaced stationary eigenstates of the metasurface. The eigenstates of the transmission matrix in Eq. (2) are polarization states that can be repre-sented as polarization ellipses with ellipticity angle χ and orientation angle ψ. Polarization states are plotted on a Poincaré sphere for easy visualization.

Simulation and experimental data in Figure 3A show how the polarization eigenstates of the metas-urface evolve as the parameters [ω, Gxy(S)] are encir-cled around the EP. On the Poincaré sphere, we plot the

eigenstates of the system found from simulations for SRR separations 5 and 6 μm and frequency range 0.42–0.44 THz. Each parameter gives two eigenstates located on opposite sides of the sphere with respect to the north pole. Following the states along the parameter loop (ω, S) → (0.42, 5) → (0.44, 5) → (0.44, 6) → (0.42, 6) → (0.42, 5), we find that each state moves halfway around the north pole, resulting in a swapping of eigenstates. The polari-zation ellipses for the eigenstates along the encircling path are explicitly shown. States from experimental data are also plotted for SRR separations 5 and 6 μm and for a frequency range 0.45–0.47 THz. We find that the states from experimental data also result in a swapping of eigen-states under encircling of the EP. Although we should be able to observe an accumulation of geometric phases on one of the states, for our current setup, it cannot be meas-ured. This is because measurements for different coupling values are conducted separately, thus making the phase relation between measurements ill-defined.

2.3.3 Asymmetric transmission of circularly polarized light

Due to the existence of a single transmission eigenstate at the EP, we expect that transmission should be asym-metric for right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) states when near the EP, which are associated with an abrupt change in phase. In general, the eigenstates of the system are given by |ψ1⟩ = α | L⟩ + β | R⟩ and |ψ2⟩ = α | L⟩ – β | R⟩. When the system is near the EP, both eigenstates are close to the |L⟩ state. Therefore, we can assume that |α |  | β |. As T | ψi⟩ = λi | ψi⟩, we find the transmission behavior for the RCP and LCP states to be

1 2

1 2| | ( ) |2 2

T L L Rλ λ β

λ λα

+⟩ = ⟩ + − ⟩ (5a)

1 2

1 2| ( ) | | .2 2

T R L Rλ λα

λ λβ

+⟩ = − ⟩ + ⟩ (5b)

From Eqs. (5a) and (5b), we can expect that the direct transmission of RCP and LCP are identical. However, for cross-polarization conversion, the conversion from RCP to LCP is larger than the conversion from LCP to RCP by a factor of (α/β)2. As we have |α |  | β |  near the EP, we expect to see an asymmetry in the transmission compo-nents TRL and TLR as the EP is approached.

We verify these predictions through simulation and experimental results for the transmission spectrum of the

1036 S.H. Park et al.: Observation of an exceptional point in a non-Hermitian metasurface

metasurface. Figure 3B shows that the conversion from LCP to RCP (TRL) is significantly lower than the conversion from RCP to LCP (TLR) when the system is near the EP (S = 6 μm). Physically, this can be understood from the fact that, near the EP, the amplitude of the RCP state is much smaller than the amplitude of the LCP state. Hence, it is relatively improbable for conversion from a high amplitude state to a low amplitude state to occur compared to conversion in the opposite direction. As the LCP state dominates more as the system approaches the EP, we expect the transmission asymmetry to also get larger. We notice that there is a mis-match of about 0.03 THz in the resonance peak between the simulation and experimental data. This discrepancy most likely stems from a difference in the refractive index

of the silicon wafer substrate. Although we do not dismiss the discrepancy as trivial, we would like to point out that it is not a major issue in observing an EP. The shift in reso-nance frequency will simply result in the EP also being shifted in the parameter space. One can see that as the system approaches the EP, TRL reaches almost zero and an abrupt jump in the phase of TRL is observed despite a small change of coupling strength between two SRRs (Figure 3C). The sensitivity in phase dispersion for TRL originates from the sensitivity of eigenvalues and eigenstates near the EP [15–18]. From Eq. (5), we find that the phase of TRL depends on β/α and λ1 – λ2. As β/α gives the eigenstate position on the Poincaré sphere and λ1 – λ2 is the eigenvalue splitting, we see that sensitivity of the eigenvalues and eigenstates

θ

k

1.0

0.8

–0.2

–0.4

–0.6

–0.8

1.2

1.4

1.6

Frequency (THz)

0.42 0.43 0.44 0.45 0.46

0.450.44 0.46 0.47 0.48 0.450.44 0.46 0.47 0.48

0.42 0.43 0.44 0.45 0.46

0.5

0.6

0.7

0.8A

B

0.5

0.6

0.7

0.8

Frequency (THz)

Eig

entr

ansm

issi

on m

agni

tude

Eig

entr

ansm

issi

on p

hase

Pha

se T

RL

(rad

.)

Exp.

Sim.

Exp.

Sim.

θ (deg.)

0.001

0.01

0.1

1

0.001

0.01

0.1

1

Frequency (THz)

40 41 42 43

0.38 0.40 0.42 0.44 0.46 0.48 0.500.38 0.40 0.42 0.44 0.46 0.48 0.50

–4

0

4

–4

0

4

Frequency (THz)

Exp.

Sim.

Exp.

Sim.

41 42

θ (deg.)

0.420.40 0.44 0.46 0.48 0.50 0.52 0.420.40 0.44 0.46 0.48 0.50 0.52

|TR

L|

Figure 4: Eigenvalue surface topology and transmission phase singularity demonstrated by varying the incident angle of radiation.(A) The topology of the self-intersecting Riemann surface is observed in the [ω, gy(θ)] parameter space. Just as we have observed by varying Gxy, we see that the crossing behavior of the eigenvalue magnitude and phase are interchanged as gy passes through the EP. From simulation and experimental results, we find that the EP located between θ = 41° and 42°. (B) We observe an abrupt phase flip in TRL as θ is varied through the EP.

S.H. Park et al.: Observation of an exceptional point in a non-Hermitian metasurface 1037

directly translate into sensitivity of the TRL phase. This rapid phase dispersion can be used for extremely sensi-tive biochemical detection [24–26].

2.3.4 Modulation of the incident angle of radiation

Up to this point, we have shown that an EP can be observed in our metasurface design through variation of the incident radiation frequency and separation between SRRs. Although this provides a straightforward and intuitive path to observing the EP, having to repeatedly fabricate new samples to change the SRR separation is a definite drawback. Repeated fabrication will inadvert-ently add fabrication errors, making it difficult to fine-tune parameters. When parameters are near the EP, errors are further amplified because the eigenvalues exhibit an enhanced sensitivity to perturbations [15, 16]. Moreover, we are also unable to perform dynamic encircling of the EP, which gives rise to phenomena such as time-asymmet-ric loops around the EP [27].

To overcome these limitations, we modulate the para-meter gy instead of Gxy by controlling the angle of incident radiation. By increasing the incident angle, we leave the in-plane x-component of the field unchanged while weak-ening the in-plane y-component and hence decreasing gy. Using this method, we are capable of probing the para-meter space with a single metasurface sample, freeing us from the limitations pointed out above.

We now show that the EP is observable in the [ω, gy(θ)] parameter space. In Figure 4A, the simulation and experi-mental data for the complex eigenvalues are shown. Just as we have seen by modulating the coupling strength, the magnitude (phase) of the eigenvalues change from cross-ing (anti-crossing) to anti-crossing (crossing) as the inci-dent angle is modulated through the EP. This implies that the self-intersecting Riemann surface topology is also cap-tured in the [ω, gy(θ)] parameter space. Figure 4B shows the magnitude and phase of the TRL transmission matrix component. Once again, as expected from the properties of an EP, we see that the magnitude of TRL decreases as the EP is approached accompanied with an abrupt phase jump.

3 Conclusion

In this work, we have designed a non-Hermitian meta-surface composed of orthogonally oriented SRRs to observe an EP. By first controlling the radiation frequency and coupling between SRR elements, we have observed

phenomena stemming from an EP, such as unique level crossing behavior, swapping of eigenstates under encir-clement of the EP, asymmetric transmission of circularly polarized light, and an abrupt phase flip in the cross-polarization transmission. This abrupt phase change for circularly polarized light could have applications in devices, including active polarization modulator and sensitive biosensors. Our work also highlights that the EP can be observed without repeated fabrication of the meta-surface by simply varying the incident angle of radia-tion, making it possible to observe phenomena related to dynamically encircling the EP. It would help to precisely locate the optimized operation point even after sample fabrication. By showing that an EP is observable in the polarization space of light, we open up a new platform for not only studying non-Hermitian physics but also devel-oping very sensitive THz systems with nontrivial phase response.

Acknowledgments: This work was supported by the Insti-tute for Basic Science of Korea (grant IBS-R011-D1, Funder Id: http://dx.doi.org/10.13039/501100010446), the National Research Foundation of Korea (NRF) through the Gov-ernment of Korea (MSIP; grants NRF-2017R1A2B3012364, Funder Id: http://dx.doi.org/10.13039/501100003621, 2017M3C1A3013923, and 2015R1A5A6001948), and the Center for Advanced Meta-Materials funded by the Gov-ernment of Korea (MSIP) as a Global Frontier Project (grant NRF-2014M3A6B3063709).

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